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A340852
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Numbers that can be factored in such a way that every factor is a divisor of the number of factors.
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14
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1, 4, 16, 27, 32, 64, 96, 128, 144, 192, 216, 256, 288, 324, 432, 486, 512, 576, 648, 729, 864, 972, 1024, 1296, 1458, 1728, 1944, 2048, 2560, 2592, 2916, 3125, 3888, 4096, 5120, 5184, 5832, 6144, 6400, 7776, 8192, 9216, 11664, 12288, 12800, 13824, 15552
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OFFSET
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1,2
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COMMENTS
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Also numbers that can be factored in such a way that the length is divisible by the least common multiple.
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LINKS
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EXAMPLE
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The sequence of terms together with their prime indices begins:
1: {}
4: {1,1}
16: {1,1,1,1}
27: {2,2,2}
32: {1,1,1,1,1}
64: {1,1,1,1,1,1}
96: {1,1,1,1,1,2}
128: {1,1,1,1,1,1,1}
144: {1,1,1,1,2,2}
192: {1,1,1,1,1,1,2}
216: {1,1,1,2,2,2}
256: {1,1,1,1,1,1,1,1}
288: {1,1,1,1,1,2,2}
324: {1,1,2,2,2,2}
432: {1,1,1,1,2,2,2}
For example, 24576 has three suitable factorizations:
(2*2*2*2*2*2*2*2*2*2*2*12)
(2*2*2*2*2*2*2*2*2*2*4*6)
(2*2*2*2*2*2*2*2*2*3*4*4)
so is in the sequence.
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MATHEMATICA
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facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[1000], Select[facs[#], And@@IntegerQ/@(Length[#]/#)&]!={}&]
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CROSSREFS
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These factorizations are counted by A340851.
A143773 counts partitions whose parts are multiples of the number of parts.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
- Factorizations -
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even numbers, even-length case A340786.
Cf. A050320, A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340609, A340654, A340655, A340827, A340830.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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